2023/05/12 by Naoya Ando, Ando, Naoya, Masaaki Umehara +1 · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research
paper · pdf · doi:10.48550/arxiv.2305.07196
In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz-Minkowski space L3. In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in L3 is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer m, there exists a germ of a space-like surface with an isolated C∞-umbilic (resp. C1-umbilic) of index (3-m)/2 (resp. 1+m/2). We also show that the indices of isolated umbilics of time-like surfaces in L3 that are not the accumulation points ofquasi-umbilics are always equal to zero. On the other hand, when quasi-umbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.